Recall now the notion of a punctured log curve, defined in [1]}. For our purposes, we will mostly think of them as ordinary log curves \(C\) , equipped with an enlarged fine log structure such that after applying functorial tropicalization to the induced morphisms of log schemes, we get a puncturing of \(\Sigma (C)\) . Most examples of families of tropical curves considered in this paper come from tropicalizing a family of punctured log curve \(C \rightarrow S\)  with target a fine and saturated log smooth log scheme \(X\) , or its Artin fan \(\mathcal {X}\) . More precisely, given a diagram in fine log schemes:
\(\begin{tikzcd}C [dd,bend right =40] \\C^\circ {u} {r} {d} & X \\S\end{tikzcd}\) 
